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Theorem isstruct2r 11955
Description: The property of being a structure with components in (1st𝑋)...(2nd𝑋). (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 18-Jan-2023.)
Assertion
Ref Expression
isstruct2r (((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅})) ∧ (𝐹𝑉 ∧ dom 𝐹 ⊆ (...‘𝑋))) → 𝐹 Struct 𝑋)

Proof of Theorem isstruct2r
Dummy variables 𝑥 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpll 518 . 2 (((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅})) ∧ (𝐹𝑉 ∧ dom 𝐹 ⊆ (...‘𝑋))) → 𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)))
2 simplr 519 . 2 (((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅})) ∧ (𝐹𝑉 ∧ dom 𝐹 ⊆ (...‘𝑋))) → Fun (𝐹 ∖ {∅}))
3 simprr 521 . 2 (((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅})) ∧ (𝐹𝑉 ∧ dom 𝐹 ⊆ (...‘𝑋))) → dom 𝐹 ⊆ (...‘𝑋))
4 simprl 520 . . . 4 (((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅})) ∧ (𝐹𝑉 ∧ dom 𝐹 ⊆ (...‘𝑋))) → 𝐹𝑉)
54elexd 2694 . . 3 (((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅})) ∧ (𝐹𝑉 ∧ dom 𝐹 ⊆ (...‘𝑋))) → 𝐹 ∈ V)
6 elex 2692 . . . 4 (𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) → 𝑋 ∈ V)
76ad2antrr 479 . . 3 (((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅})) ∧ (𝐹𝑉 ∧ dom 𝐹 ⊆ (...‘𝑋))) → 𝑋 ∈ V)
8 simpr 109 . . . . . 6 ((𝑓 = 𝐹𝑥 = 𝑋) → 𝑥 = 𝑋)
98eleq1d 2206 . . . . 5 ((𝑓 = 𝐹𝑥 = 𝑋) → (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ↔ 𝑋 ∈ ( ≤ ∩ (ℕ × ℕ))))
10 simpl 108 . . . . . . 7 ((𝑓 = 𝐹𝑥 = 𝑋) → 𝑓 = 𝐹)
1110difeq1d 3188 . . . . . 6 ((𝑓 = 𝐹𝑥 = 𝑋) → (𝑓 ∖ {∅}) = (𝐹 ∖ {∅}))
1211funeqd 5140 . . . . 5 ((𝑓 = 𝐹𝑥 = 𝑋) → (Fun (𝑓 ∖ {∅}) ↔ Fun (𝐹 ∖ {∅})))
1310dmeqd 4736 . . . . . 6 ((𝑓 = 𝐹𝑥 = 𝑋) → dom 𝑓 = dom 𝐹)
148fveq2d 5418 . . . . . 6 ((𝑓 = 𝐹𝑥 = 𝑋) → (...‘𝑥) = (...‘𝑋))
1513, 14sseq12d 3123 . . . . 5 ((𝑓 = 𝐹𝑥 = 𝑋) → (dom 𝑓 ⊆ (...‘𝑥) ↔ dom 𝐹 ⊆ (...‘𝑋)))
169, 12, 153anbi123d 1290 . . . 4 ((𝑓 = 𝐹𝑥 = 𝑋) → ((𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥)) ↔ (𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋))))
17 df-struct 11946 . . . 4 Struct = {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}
1816, 17brabga 4181 . . 3 ((𝐹 ∈ V ∧ 𝑋 ∈ V) → (𝐹 Struct 𝑋 ↔ (𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋))))
195, 7, 18syl2anc 408 . 2 (((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅})) ∧ (𝐹𝑉 ∧ dom 𝐹 ⊆ (...‘𝑋))) → (𝐹 Struct 𝑋 ↔ (𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋))))
201, 2, 3, 19mpbir3and 1164 1 (((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅})) ∧ (𝐹𝑉 ∧ dom 𝐹 ⊆ (...‘𝑋))) → 𝐹 Struct 𝑋)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  w3a 962   = wceq 1331  wcel 1480  Vcvv 2681  cdif 3063  cin 3065  wss 3066  c0 3358  {csn 3522   class class class wbr 3924   × cxp 4532  dom cdm 4534  Fun wfun 5112  cfv 5118  cle 7794  cn 8713  ...cfz 9783   Struct cstr 11940
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-rex 2420  df-rab 2423  df-v 2683  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-iota 5083  df-fun 5120  df-fv 5126  df-struct 11946
This theorem is referenced by:  isstructr  11959
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